"The bill will pass. I think we're on the verge of getting 70 votes. That is my goal," Graham said on "Fox News Sunday." "We're very, very close to 70 votes. The Hoeven-Corker amendment I think gets us over the top."

Graham, a member of the bipartisan Gang of Eight, called the Corker-Hoeven amendment "the most aggressive attempt to control the Southern border and regain our sovereignty."

He stressed that Republicans will be blamed if the bill fails to pass the Senate, which would continue to drive Latinos away from the party. The GOP is seeking to court more Latino voters after an overwhelming majority of them voted for President Obama in the 2012 election.

"If it fails, then we are blamed for its failure," Graham said. "Our party is in trouble with Hispanics, not because we're conservative, but because of the rhetoric and the way that we've handled this issue."

"I want to get reattached to the Hispanic community…pass comprehensive immigration reform, grow this party," he said. "The Hispanic community is very close to our values, but we have driven them away over this issue. Let's fix this problem for the good of the country and for the good of the party, and this bill does that."

"What's making this contentious, what's making it politically divisive, is the Senate Gang of Eight has insisted on wrapping all of these issues up into one 1,200 page bill," Lee said also appearing on "Fox News Sunday." "They're putting the single most controversial…issue, that of amnesty and ultimate citizenship, into a 1,200 page bill."

Lee also said the Senate bill lacked teeth on border enforcement and few members have had the chance to fully weigh the merits of the Corker-Hoeven amendment. However, he predicted the bill will ultimately pass the Senate.

"They told us [the bill] would be tough but fair, and it's neither. It's not tough on those who have broken the law and it's not fair for those who have been patiently waiting their turn in line to come to this country legally," Lee said.

"Look, we all want immigration reform, but this bill is not the vehicle for getting there," he said.